The second modification applies the sampling technique from the -literature. This technique was first introduced for 3-D in [1]} (who called it the “averaging maneuver”), and later adapted to high dimensions by [2]}. The basic idea is that instead of having a single \(x\) -team we introduce many copies for the \(x\) -team. Each team tries to imitate a different shift of the \(y\) -team. We design the shits to ensure that if one \(x\) -team is located close to the \(2^{-N_{in}}\) -grid then the other teams necessarily are located far from the grid.
